General reference is made to S. Haykin, Commnunications Systems, (3.sup.rd Ed. New York, John Wiley & Sons, 1994) for a detailed teaching of angle-modulation systems. Angle-demodulation can be either non-coherent or coherent.
Non-coherent Angle-demodulation
Referring to FIG. 1, typically, non-coherent angle-modulation recovery or estimation uses a band-pass filter 101, a phase differentiator 102, and a low-pass filter/integrator 103. Band-pass filter 101 pre-conditions an input signal by passing through the frequency band occupied by the desired signal, s(t), and rejecting all other frequencies. However, since some of the distortion, W(t), occupies the same frequency band as s(t), band-pass filter 101 passes both the desired signal, s(t), and the distortion, N(t). The output of band-pass filter 101 can thus be modeled as EQU X(t)={s(t)}.sub.BPF +N(t),
where {.multidot.}BPF represents the effects of band-pass filter 101 and N(t)={W(t)}.sub.BPF. This filtered IF signal, X(t), is then input to a phase differentiator 102. The output is .phi..sub.1 '(t), as shown.
Among the many ways to carry out this task are (1) zero crossing measurement (which is inversely related to signal frequency) and (2) frequency-to-voltage conversion with envelope detection (often referred to as an FM discriminator). Either measures the rate-of-change of the phase of a signal with respect to time. The signal is processed by a low-pass filter within low-pass filter/integrator 103, whose cut-off frequency is commensurate with the bandwidth of the original message or information signal, m(t). The variable of interest, either .phi.'(t), the low-pass filtered rate-of-change of phase measurement, or its integral, ##EQU1## depends on the type of angle modulation, frequency or phase. For frequency modulation, .phi.(t) is generated at the transmitter source as ##EQU2## in which case the output signal, .phi.'(t).congruent.2.pi..multidot..DELTA..function..multidot.m.sub.FM (t), is proportional to the original message, m.sub.FM (t). Generally, the message signal, m(t)=m.sub.FM (t) or m(t)=m.sub.PM (t), is normalized such that -1.ltoreq.m(t).ltoreq.+1, and the factor .DELTA..function. controls the frequency deviation of s(t). For phase modulation, .phi.(t) is generated at the transmitter source as EQU .phi.(t)=k.sub.p .multidot.m.sub.PM (t),
and the output is .phi.(t).congruent.k.sub.p .multidot.m.sub.PM (t), where k.sub.p is a proportionality scale factor. More sophisticated angle-modulation systems may employ pre-emphasis/de-emphasis filters that operate on the message signal and the estimated message respectively to help offset the adverse effects of noise. However, without loss of generality, these filters can be modeled as incorporated into our message signal and low-pass filter/integrator 103.
To employ analytic signal representation in any practical system, the signal processed must be a band-pass signal. FIG. 2 illustrates an equivalent model of phase differentiator 102. An analytic signal extractor 201 generates an analytic signal, X.sub.+ (t). This signal contains both the imposed modulation and distortion. It can be represented as EQU X.sub.+ (t)=.vertline.A(t).vertline.exp{j.multidot.[2.pi..function..sub.c t+.phi.(t)+.eta.(t)-.theta.]}.
Here .vertline.A(t).vertline. is the envelope of X.sub.+ (t), and .theta. is a constant phase offset. The term .function..sub.c represents the center frequency, which is assumed to be known within a small tuning error. The term .eta.(t) represents a phase angle distortion from the imperfect receive channel. Following analytic signal extractor 201, a frequency translation module 202 multiplies the input, X.sub.+ (t), by the complex exponential EQU z.sub.- (t)=exp{-j.multidot.2.pi..function..sub.c t},
where .function..sub.c is the known constituent of the center frequency. This complex product produces the complex envelope of X(t). By employing analytic signal notation, the input signal becomes, in complex envelope form EQU X(t)=.vertline.a(t).vertline.exp{j.multidot.[2.pi..function..sub.e t+.phi.(t)+.eta.(t)-.theta.]},
where .function..sub.e =.function..sub.c -.function..sub.c is a small error in tuning frequency. Alternatively, X(t) can be represented in rectangular form as EQU X(t)=X.sub.i (t)+jX.sub.q (t),
where X.sub.i (t)=.vertline.A(t).vertline.cos[2.pi..function..sub.e t+.phi.(t)+.eta.(t)-.theta.] is the real and X.sub.q (t)=.vertline.A(t).vertline.sin[2.pi..function..sub.e t+.phi.(t)+.eta.(t)-.theta.] the imaginary component of X(t). The rate-of-change of phase of the complex signal X(t), after processing through a rate-of-change of phase module 203, is ##EQU3## From the above, the rate-of-change of phase of X(t) is EQU .phi..sub.1 '(t)=2.pi..function..sub.e +.phi.'(t)+.eta.'(t) (radians per second).
This estimate contains a bias term, 2.pi..function..sub.e, and an error term, .eta.'(t). It can easily be converted to units of Hertz by scaling with the factor 1/2.pi.. We find that for frequency modulation, with the low-pass filtering process indicated as {.multidot.}.sub.LPF, EQU .phi.'(t)={.phi..sub.1 '(t)}.sub.LPF ={2.pi..function..sub.e +2.pi..DELTA..function..multidot.m.sub.FM (t)+.eta.'(t)}.sub.LPF.
For phase modulation, EQU .phi.(t)={2.pi..function..sub.e t+k.sub.p .multidot.m.sub.PM (t)+.eta.(t)+.theta..sub.c }.sub.LPF,
where .theta..sub.c is a constant phase offset that depends on initial conditions. Note that m.sub.FM (t) and m.sub.PM (t) are related as ##EQU4##
The above summarizes non-coherent angle-demodulation. Each of the many implementations of the method can be modeled as shown. Note that these implementations include non-coherent angle-demodulation techniques that perform, prior to angle-modulation recovery, some type of "hard limiting", or normalization, on the envelope of X(t). This normalization is made evident by the factor 1/(X.sub.i.sup.2 (t)+X.sub.q.sup.2 (t)) employed in the phase differentiation process. The purpose of normalization is to desensitize the phase angle measurement to variations in envelope, thus reducing the phase distortion component, .pi.(t).
The development above exposes the limitations of the non-coherent method of angle-demodulation. This prior-art method does not take advantage of the fact that the message signal, m(t), and therefore the modulated signal, EQU s.sub.+ (t)=.vertline.a(t).vertline.exp{j.multidot.[2.pi..function..sub.c t+.phi.(t)-.theta.]},
can be highly correlated at consecutive instants. (Here, .vertline.a(t).vertline. is a slowly changing or constant envelope present on the transmitted signal.) More specifically, the spectral content of m.sub.FM (t) is essentially limited to some maximum frequency, .function..sub.m Hz. Therefore, when the deviation ratio .beta.=.DELTA..function./.function..sub.m is large (i.e., greater than 10), it is possible to employ adaptive band-pass filtering prior to measuring rate-of-change of phase. By doing so, the bandwidth of the band-pass filter process can be narrowed, thereby rejecting more of the additive distortion, W.sub.+ (t), the analytic constituent of W(t).
An additional related disadvantage of the prior-art, non-coherent method of angle-demodulation arises from the tuning error, .function..sub.e. This tuning offset can lead to an increase in the distortion, .eta.(t), since band-pass filter 101 operates at the estimated center frequency, .function..sub.c. The resulting off-centering of the input signal causes a distortion of the desired signal, so that {s(t)}.sub.BPF .noteq.s(t). In particular, the band-pass filter will have undesired attenuation and phase changes near the band edges that affect adversely the angle-demodulation process.
FIGS. 3(a) and 3(b) demonstrate this tuning error. FIG. 3(a) shows correct tuning frequency; FIG. 3(b), tuning error offset. Referring to FIGS. 3(a) and 3(b), dashed lines represent a magnitude response 301 of band-pass filter 101. The corners of the dashed lines show the lower and upper cut-off frequencies of band-pass filter 101. Solid lines represent a modulated signal spectrum 302.
Referring to FIG. 3(a), band-pass filter 101 causes little or no distortion, since modulated signal spectrum 302 falls entirely within the pass-band between the cut-off frequencies. Referring to FIG. 3(b), because modulated signal spectrum 302 goes beyond the upper cut-off frequency, magnitude response 301 of band-pass filter 101 distorts modulated signal spectrum 302.
Coherent Angle-demodulation
L. H. Enloe, "Decreasing the Threshold in FM by Frequency Feedback," 50 Proc, IRE 18-30 (January 1962) provides an overview of coherent angle-demodulation and addresses its limitations and disadvantages. The advantage of coherent over non-coherent angle-demodulation methods is that coherent methods utilize the a priori knowledge that a signal has a large modulation index, .beta.. Coherent methods are therefore able to reject more of the additive distortion, W.sub.+ (t), while minimizing the rejection of the desired signal, s.sub.+ (t).
The prior art teaches two specific coherent angle-demodulation methods in particular: phase lock loop (PLL) and FM with feedback (FMFB). There is no consensus which device performs better. J. A. Develet Jr., "Statistical Design and Performance of High-Sensitivity Frequency Feedback Receivers," IEEE Trans. On Military Electronics 281-284 (October 1963), identifies PLL and FMFB devices as "equivalent servo-mechanisms" under reasonable input signal and loop conditions. This discussion therefore presents only the FMFB demodulator.
Referring to FIG. 4, the FMFB demodulator employs low-pass filtering and integration within the device itself. However, further low-pass filtering and integration processes can be performed externally to the FMFB demodulator. Thus the FMFB demodulator replaces phase differentiator 102 of FIG. 1. If the negative of the derivative of the angle of the unit-envelope prediction signal, s.sub.+ *(t), closely follows the derivative of the angle of the modulated constituent of input signal, X.sub.+ (t), then the output of a complex multiplier 400 EQU X.sub.e (t)=X.sub.+ (t).multidot.s.sub.+ *(t)
contains a signal constituent with a reduced modulation index. This message-bearing constituent of the signal, X.sub.e (t), can pass through a band-pass filter 401, which is narrower than band-pass filter 101. For N.sub.+ (t), the analytic constituent of the distortion component, when N.sub.+ (t) and s.sub.+ (t) are not highly correlated and s.sub.+ (t) has sufficient strength, band-pass filter 401 passes less of N.sub.+ (t).
As a practical matter, band-pass filter 401 is a pair of identical real-valued low-pass filters, each operating on the real and imaginary components of X.sub.e (t). Thus the input and output of band-pass filter 401 are complex. Since band-pass filter 401 operates at an IF of 0 Hz, both the input and output are complex envelope signals. As such, a rate of-change of phase module 402 can be implemented as previously described. Passing this result to a low-pass filter/integrator 403 results in the reduced index frequency modulation estimate, .phi..sub.FB '(t). Given that the goal is to reduce as far as possible the modulation index at the output of band-pass filter 401, .phi..sub.FB '(t) can be viewed as an error signal, which should approach some small level. By integrating this "error signal" to obtain .phi..sub.FB (t) and by controlling sensitivity with a feedback gain 404, K, we can maintain a good quality prediction signal, s.sub.+ *(t). A negating angle-modulator 405 simply generates the unit-envelope prediction signal, EQU s.sub.+ *(t)=exp{-j.multidot.[2.pi..function..sub.c t+K.phi..sub.FB (t]},
which reduces the modulation index of s.sub.+ (t). Thus negative feedback is employed through the phase of the prediction signal, s.sub.+ *(t). With proper choice of feedback gain 404, band-pass filter 401, and low-pass filter/integrator 403, the FMFB system remains stable and reliably demodulates the input, X.sub.+ (t).
Though there are limitations to coherent angle-demodulation, both FMFB and PLL demodulators reduce the distortion, .eta.(t), in large .beta. systems. Specifically, these methods reduce the FM threshold effect, where a rapid decrease in output signal-to-noise ratio ("SNR") occurs for small decreases in input SNR. This threshold occurs at or about 10 dB input SNR. Threshold improvements from 3 to 7 dB or more have been reported. Another advantage of the coherent method is that the adverse effects of a tuning error, .function..sub.e, can be mitigated, as coherent angle-demodulation tracks the center frequency, .function..sub.c, thereby providing automatic tuning.
Coherent angle-demodulation has disadvantages. In a system with large .beta., processing with a particular combination of feedback gain 404, band-pass filter 401, and low-pass filter/integrator 403 can reduce the noise substantially. However, such processing can also cause excessive distortion of the original modulation. The coherent angle-demodulator provides no mechanism to compensate for this distortion.
Another disadvantage of the coherent angle-demodulator is that the modulation index is reduced. While a lower modulation index allows band-pass filter 401 to reduce the additive distortion, N(t), there is a commensurate reduction in the strength of the recovered modulation signal. The result is that coherent methods perform essentially the same as non-coherent methods of angle modulation recovery at input SNR values above threshold.
Standard Angle-Demodulation Results
Referring to FIG. 5, simulation results are presented for the FM demodulation of a pulsed carrier with imposed frequency modulation to demonstrate typical angle-demodulation. A discrete-time implementation employed the common rate, F.sub.s, of two samples per second. (This arbitrary but convenient choice of sample rate for simulations leads to a Nyquist bandwidth of 1 Hz.) The results come from a backward-difference FM demodulator with no explicit low-pass post filtering.
This demodulator is described in detail by the present inventor in "Numerical FM Demodulation Enhancements," Rome Laboratory Technical Report RL-TR-96-91, June 1996. The imposed FM modulation was a linearly varying instantaneous frequency, from approximately -0.08 radians, to +0.08 radians. Additive white Gaussian noise ("AWGN") was combined with the pre-pulsed carrier at a SNR of 30 dB. Pulses were 200 samples in duration. A total of 100 pulses were generated, band-pass filtered, demodulated, and the results averaged for comparison to the actual imposed FM modulation.
The band-pass filter was implemented in complex form with a finite-impulse response ("FIR") filter of 129 coefficients resulting from the Hanning windowed method of FIR filter design. A cutoff frequency of 0.25/4 Hz was used.
Referring to FIG. 6, the difference between the measured and actual FM modulation was also determined. The corresponding root-mean-square ("RIMS") error was 0.015182 radians.
The previous discussion demonstrates the drawbacks of the prior art. What is needed is an improved system and method to estimate the angle-modulation imposed on a transmitted RF or IF carrier, especially when communications channels add distortions such as noise to transmitted signals.
All of the references cited herein are incorporated by reference in their entireties for their teachings.